BUS606: Operations and Supply Chain Management

Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving average (MA) models.

These three classes depend linearly on previous data points. Combinations of these three types produce autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) models. The autoregressive fractionally integrated moving average (ARFIMA) model generalizes the former two. Extensions of these classes to deal with vector-valued data are available under the heading of multivariate time-series models. An additional set of extensions of these models is available for use where the observed time-series is driven by some "forcing" time-series "(which may not have a causal effect on the observed series): the distinction from the multivariate case is that the forcing series may be deterministic or under the experimenter's control. For these models, the acronyms are extended with a final "X" for "exogenous".

Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models over those from linear models.When dealing with nonlinearities, Campbell, Lo, and MacKinlay make the distinction between: (1) Linear time series where shocks are assumed to be uncorrelated but not necessarily identically independent distributed (iid), and (2) Nonlinear time series where shocks are assumed to be iid, but there is a nonlinear function relating the observed time series $\left\{x_{t}\right\}$ and the underlying shocks.

Among the most applied non-linear time series models in financial data are those representing changes of variance along time (heteroskedasticity). These models are called autoregressive conditional heteroskedasticity (ARCH) and the collection comprises a wide variety of representation (GARCH, TARCH, EGARCH, FIGARCH, CGARCH, etc). Here changes in variability are related to, or predicted by, recent past values of the observed series. This is in contrast to other possible representations of locally varying variability, where the variability might be modeled as being driven by a separate time-varying process, as in a doubly stochastic model.